The Lagrange problem on an optimal column: old and new results

The paper is devoted to the optimization and post-buckling behavior of columns elastically supported at both ends. The unimodal solutions are analyzed, and it is shown that for nonzero support stiffnesses they are not optimal. The bimodal formulation of the problem is set up. By using analytical expressions for bimodal columns obtained earlier, the bimodal optimal solutions are integrated for different values of the support stiffnesses. With the assumption of geometrical nonlinearity, the post-buckling behavior of the bimodal optimal columns is studied. It is shown that the initial post-buckling behavior is governed by four supercritical solutions emanating from the trivial equilibrium state at the critical load. The stability of the new equilibrium states is investigated by using the second variation of the total potential energy. It is shown that only two post-buckling equilibrium states are stable while the other two are unstable, this conclusion being valid for all considered values of the support stiffnesses. An important limit case of a clamped–simply supported column that has caused debate in many publications is analyzed.